Consider the hyperbola $H: x^{2} - y^{2} = 1$ and a circle $S$ with center $N(x_{2}, 0)$. Suppose that $H$ and $S$ touch each other at a point $P(x_{1}, y_{1})$ with $x_{1} > 1$ and $y_{1} > 0$. The common tangent to $H$ and $S$ at $P$ intersects the x-axis at point $M$. If $(l, m)$ is the centroid of the triangle $PMN$, then the correct expression(s) is(are)

Consider the hyperbola H: x² - y² = 1 and a circle S with center N(x₂, 0). Suppose that H and S touch each other at a point P(x₁, y₁) with x₁ > 1 and y₁ > 0. The common tangent to H and S at P intersects the x-axis at point M. If (l, m) is the centroid of the triangle PMN, then the correct expression(s) is(are)

$H: x^2 - y^2 = 1, \quad S: \text{circle, center } N(x_2,0)$

$H \text{ and } S \text{ touch at } P(x_1,y_1), \quad x_1>1,\ y_1>0$

$\text{common tangent at } P \text{ meets } x\text{-axis at } M$

centroid of triangle PMN = (l,m)

$\text{To find: which of the given derivatives are correct}$

$H: x^2 - y^2 = 1 \quad (\text{asymptotes } y=\pm x)$

$P(x_1,y_1) \in H \;\Rightarrow\; x_1^2 - y_1^2 = 1$

$1.\ \text{tangent at } P \Rightarrow M$

$2.\ \text{touch condition} \Rightarrow N$

$3.\ \text{centroid } (l,m) \Rightarrow \text{check derivatives}$

$\text{tangent to } H \text{ at } P: \quad x x_1 - y y_1 = 1$

$y=0 \;\Rightarrow\; x x_1 = 1 \;\Rightarrow\; x = \dfrac{1}{x_1}$

$M\left(\dfrac{1}{x_1},0\right)$

$NP \perp \text{tangent} \;\Rightarrow\; (x_1-x_2,\ y_1) \parallel (x_1,\ -y_1)$

$\dfrac{x_1-x_2}{x_1} = \dfrac{y_1}{-y_1} = -1$

$x_2 = 2x_1 \;\Rightarrow\; N(2x_1,0)$

$l = \dfrac{x_1+\frac{1}{x_1}+2x_1}{3} = x_1+\dfrac{1}{3x_1}$

$m = \dfrac{y_1+0+0}{3} = \dfrac{y_1}{3}$

$\dfrac{dl}{dx_1} = 1 - \dfrac{1}{3x_1^2} \quad \checkmark$

$\text{option (1) is True}$

$\dfrac{dl}{dx_1} = 1 - \dfrac{1}{3x_1^2} \;\neq\; 1 + \dfrac{1}{3x_1^2}$

$\text{option (3) is False}$

$y_1=\sqrt{x_1^2-1} \;\Rightarrow\; \dfrac{dy_1}{dx_1} = \dfrac{x_1}{\sqrt{x_1^2-1}}$

$\dfrac{dm}{dx_1} = \dfrac{1}{3}\cdot\dfrac{x_1}{\sqrt{x_1^2-1}} \quad \checkmark$

$m = \dfrac{y_1}{3} \;\Rightarrow\; \dfrac{dm}{dy_1} = \dfrac{1}{3} \quad \checkmark$

$\text{option (4) is True}$

$\boxed{\text{Correct options: (1), (2) and (4)}}$

$\text{Answer: (1), (2), (4)}$